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Theorem ffnafv 28033
Description: A function maps to a class to which all values belong, analogous to ffnfv 5685. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
ffnafv  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B ) )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem ffnafv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ffn 5389 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
2 fafvelrn 28032 . . . 4  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( F''' x )  e.  B
)
32ralrimiva 2626 . . 3  |-  ( F : A --> B  ->  A. x  e.  A  ( F''' x )  e.  B
)
41, 3jca 518 . 2  |-  ( F : A --> B  -> 
( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
) )
5 simpl 443 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  F  Fn  A )
6 afvelrnb0 28026 . . . . 5  |-  ( F  Fn  A  ->  (
y  e.  ran  F  ->  E. x  e.  A  ( F''' x )  =  y ) )
7 nfra1 2593 . . . . . 6  |-  F/ x A. x  e.  A  ( F''' x )  e.  B
8 nfv 1605 . . . . . 6  |-  F/ x  y  e.  B
9 rsp 2603 . . . . . . 7  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( x  e.  A  ->  ( F''' x )  e.  B
) )
10 eleq1 2343 . . . . . . . 8  |-  ( ( F''' x )  =  y  ->  ( ( F''' x )  e.  B  <->  y  e.  B ) )
1110biimpcd 215 . . . . . . 7  |-  ( ( F''' x )  e.  B  ->  ( ( F''' x )  =  y  ->  y  e.  B ) )
129, 11syl6 29 . . . . . 6  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( x  e.  A  ->  ( ( F''' x )  =  y  ->  y  e.  B ) ) )
137, 8, 12rexlimd 2664 . . . . 5  |-  ( A. x  e.  A  ( F''' x )  e.  B  ->  ( E. x  e.  A  ( F''' x )  =  y  ->  y  e.  B ) )
146, 13sylan9 638 . . . 4  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  ( y  e.  ran  F  ->  y  e.  B ) )
1514ssrdv 3185 . . 3  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  ran  F  C_  B )
16 df-f 5259 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
175, 15, 16sylanbrc 645 . 2  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B
)  ->  F : A
--> B )
184, 17impbii 180 1  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   ran crn 4690    Fn wfn 5250   -->wf 5251  '''cafv 27972
This theorem is referenced by:  ffnaov  28059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-dfat 27974  df-afv 27975
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